1. Problem 2.23.I All satellites near planets have nearly circular orbits, so you can assume e=0. Before starting this problem, make a careful sketch of Eq. 2.44a vs. distance for orbits within 10 radii of Mars' surface; indicate Phobos and Deimos on your sketch (If you are using the first edition, you'll need to get Eq. 2.44a from a student with the second edition). What are the main features of your sketch and do they make sense physically? Note that a(0) is simply a(t=0). After solving the problem, calculate the impact timescale for Mars' moon Phobos. Assume kT1 = 0.14 and Q1 =86, and get other values that you need from the Satellite Calculator at http://janus.astro.umd.edu/astro/calculators/scalc.html. Comment on your numerical result.
2. This problem is motivated by problem 2.32.I, but instead of parts a) and b), derive the general result for launch from a distance r along an elliptical orbit. Show that the special cases a) and b) follow from your more general result.
3. Go to the Central Force
Integrator from the class webpage.
a) Change the defaults to
vr=0.3, B=0.05, and n=-4. This adds the General
relativistic perturbation (greatly magnified) to Newtonian
gravity. Does the orbital pericenter precess (rotate in the same
direction as the orbital motion) or regress (go in the opposite
direction)?
b) Now change the power on the perturbation force.
Which forces lead to precession and which to regression? What happens
when you change the sign and/or the magnitude of the perturbation?
c) Repeat part b) for the harmonic oscillator (m=1), subject to
various perturbation forces, changing the other parameters as
needed.
d) Generalize your observations. What is the condition on
the perturbation force that leads to precession? Print out and turn in
a few plots to support your conclusions.
e) (Optional) Want to
explore more? See which force laws allow stable circular orbits (put
vr=0 and vθ=.99999999). See if you can
find other force laws that lead to closed orbits. See if you can
create an orbit that looks like a 5-pointed star.
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